Objectives of the course.

The course aims at providing an introduction to the concepts of numerical approximation, error analysis and computational methods that represent the main constituents of Numerical Mathematics. As a motivating example to this study, we will consider an initial value/boundary value problem in one spatial dimension to physically describe the dynamical distribution of temperature and heat flux in a bar under the action of externally applied forces and of distributed sources/sinks of heat. Starting from this example, in the first part of the course the following topics will be introduced, analyzed and implemented on the computer:

1. polynomial approximation of functions and data;
2. approximation of integrals through numerical quadratures;
3. numerical solution of ordinary differential equations (ODEs);
4. numerical solution of linear algebraic systems.

In the second part of the course, we will apply the above mentioned methodologies in the context of the Galerkin Finite Element Method (GFEM) for the numerical approximation of elliptic and parabolic partial differential equations (PDEs) in one spatial dimension (1D). All theoretical concepts, methods and algorithms will be accompanied by a numerical verification on the computer during the laboratory sessions that will constitute a fundamental complement of ex-cathedra classes. Computational examples, exercises and coding will be conducted using the Matlab scientific environment.

Detailed program.

A. Fundamentals of Numerical Analysis

1. Approximation of functions and data: polynomial interpolation and least squares fitting.
2. Numerical integration using quadrature formulae: rectangle, trapezoidal, Cavalieri-Simpson and Gaussian quadratures.
3. Numerical solution of ODEs : the theta-method, zero-stability, convergence and absolute stability.
4. Numerical linear algebra: direct and iterative methods.

B. The Galerkin Finite Element Method for PDEs

1. The GFEM approximation of elliptic problems in 1D: weak formulation, algebraic formulation and error estimates.
2. The GFEM approximation of parabolic problems in 1D: weak formulation, algebraic formulation and error estimates.

The exam consists of the solution of a number of exercises (typically three) and is conducted in a computer room. Each exercise is divided into two main parts. The first part of each exercise includes theoretical questions. The second part of each exercise requires the solution of specific questions using the methods and coded algorithms that have been illustrated during the course. All computations are carried out using the Matlab scientific software environment.

The updated version of the lecture notes will be made available to students before the beginning of the course.